Inter-House Maths Olympiad

Enjoy the challenge!

Mathematics is a language, and the Book of Nature is written in this language.

Mathematics is the finest art, the most intellectual art in human history.

Mathematics is the whole world for intellectual travels and an inexhaustible treasure box with planets of ideas.

Mathematics is an intellectual sport: it is more intellectual than Chess, more rich than the game of Go.

The mathematical Olympiads are spread world-wide. Many mathematicians have played this game before, and many continue to compete with each other during their entire lives. Almost everyone can run 100 m in 30 sec. But no-one can do it in 6 sec. Many can lift 60 kg. However, even the strongest person cannot lift 300 kg above their head. You can see: the strongest body is only 5 times stronger (even less!) than an average one. But there is no limit for intellect. In intellect you can be many times stronger than many other people and this is especially clear in solving difficult problems: you can solve problems that no-one could solve before.

Sample Problems:

1. The Statue of Liberty is 46 feet tall, and stands on a platform 47 feet tall. How far from the statue along the ground should I stand to get the largest viewing angle possible? (The "viewing angle" is the angle between the line connecting my eyes to the statue's feet, and the line connecting my eyes to the statue's crown.)

2. A farmer grows bananas in a desert oasis. He has 3000 bananas and the market is 1000 miles away. He has only a camel to transport bananas, but there are two problems: (i) The camel can carry at most 1000 bananas at a time. (ii) The camel will only walk if munching on a banana, and eats one banana for every mile it walks. What is the maximum number of bananas the farmer can get to market using ONLY the camel to transport them?

3. In a contest, no student solved all problems. Each problem was solved by exactly three students and each pair of problems was solved by exactly one student. What is the maximum number of problems in this contest?

4. The function f(n) = an + b, where a and b are integers, is such that for every integer n, f(3n + 1), f(3n) + 1 and 3f(n) + 1 are three consecutive integers in some order. Determine all such f(n).

5. Martian Citizenship Quiz

The engine on the Enterprise malfunctioned and the spaceship crash landed on Mars. Repair was beyond the capability of the crew, but the Martians would render assistance only to their own citizens. Thus one of the crew members must succeed in gaining Martian citizenship.

The Martian Citizenship Quiz consisted of the same 30 true or false questions in the same order. Each question must be answered, and the number of correct responses was announced. A candidate succeeded if and only if all 30 questions were answered correctly. A failed candidate might not apply again.

Not knowing a word in the Martian language, the crew had to formulate a plan, based on the feedback from the failed attempts, so that by the time the last of the 24 crew members applied, success was guaranteed. The Enterprise could then be repaired, and they could continue on their mission. Was this possible?

6. In triangle ABC, AB = AC and angle A is 100o. D is the point on BC such that AC = DC, and F is the point on AB such that DF is parallel to AC. Determine angle DCF.

7. What is the maximum number of “Friday the 13ths” that can occur in a normal 365-day year? What is the minimum number that must occur? (Recall that each of April, June, September and November has 30 days, February has 28 days, and all the other months have 31 days.)

8. My partner and I invited 10 couples to a party at our house. I asked everyone present, except myself, how many people they shook hands with. It turns out that everyone shook hands with a different number of people. If we assume that no one shook hands with his or her partner, how many people did my partner shake hands with?

9. Two people take turns cutting up a rectangular chocolate bar which is 6 × 8 squares in size. You are allowed to cut the bar only along a division between the squares and your cut can be only a straight line. For example, you can turn the original bar into a 6 × 2 piece and a 6 × 6 piece, and this latter piece can be turned into a 1 × 6 piece and a 5 × 6 piece. When the bar has been divided into single squares, the person who made the last division wins all the chocolate. Is there a winning strategy for the first or second player? What about the general case where the starting bar is m × n squares in size?